{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $R$ be a commutative [[Noetherian ring|0670T59HGE]], let $I$ be an [[ideal|096Z9MP2ME]] of $R$, and let $\\hat R$ be the [[$I$-adic completion|NS84D2920R]] of $R$.\n\n1. For every [[finitely generated|05QY1VEBYD]] $R$-module $M$, the natural map $\\hat R \\otimes_R M \\to \\hat M$ is an isomorphism, where $\\hat M = \\varprojlim_n M/I^n M$.\n2. $\\hat R$ is a [[flat|73RYYEVDKA]] $R$-module.\n3. If $I$ is contained in the [[Jacobson radical|XA0DCNH7WY]] of $R$, then $\\hat R$ is a [[faithfully flat|C31PYYPRGC]] $R$-module; this applies when $R$ is local and $I \\neq R$.\n\n## Remarks\n\nBy the [[Artin–Rees lemma|30H6B0G2FT]], completion $M \\mapsto \\hat M$ is an exact functor on finitely generated $R$-modules.\nBoth $M \\mapsto \\hat R \\otimes_R M$ and $M \\mapsto \\hat M$ are right exact and agree on finite free modules, so they agree on finitely presented modules, which gives clause 1.\nThen for an ideal $J$ of $R$ the map $\\hat R \\otimes_R J \\to \\hat R$ is the injective map $\\hat J \\to \\hat R$, which gives clause 2 by the ideal criterion for flatness.\nFor clause 3, each maximal ideal $\\mathfrak n$ contains $I$, so $\\hat R/\\mathfrak n \\hat R \\cong R/\\mathfrak n \\neq 0$.\nClause 3 can fail without its hypothesis.\nThe $p$-adic completion $\\Z_p$ of $\\Z$ is flat over $\\Z$, but $\\Z_p \\otimes_\\Z \\Z/q\\Z = 0$ for a prime $q \\neq p$, because $q$ is a unit in $\\Z_p$.\nSo $\\Z_p$ is not faithfully flat over $\\Z$.\n\n## Explanations\n\n### L1 human\n\nCompleting a ring adds limits of approximations, like passing from integers to $p$-adic integers.\nFlatness means that this operation does not destroy injectivity of maps between modules.\nSo information about a Noetherian ring and its modules is kept faithfully when one passes to the completion, at least near the ideal used for completing.\n\n### L2 human\n\nFlatness lets one move properties between a Noetherian local ring $R$ and its completion $\\hat R$.\nFor instance $\\dim \\hat R = \\dim R$, and $R$ is regular, or Cohen–Macaulay, if and only if $\\hat R$ is.\nCombined with the [[Cohen structure theorem|N3XH42KVV3]], this reduces many questions about local rings to quotients of power series rings.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "KBN13NPYJQ",
      "73RYYEVDKA"
    ],
    "kind": "statement",
    "msc": [
      "13B35",
      "13C11",
      "13J10"
    ],
    "name": "Completion is flat",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Completion_of_a_ring"
      }
    ],
    "tag": "NPF42DJMBX",
    "uses": [
      "0670T59HGE",
      "096Z9MP2ME",
      "NS84D2920R",
      "73RYYEVDKA",
      "C31PYYPRGC",
      "XA0DCNH7WY",
      "05QY1VEBYD",
      "F0HQXQ0HCD",
      "30H6B0G2FT"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/NPF42DJMBX/",
  "links": [
    {
      "field": "about",
      "target": "KBN13NPYJQ"
    },
    {
      "field": "about",
      "target": "73RYYEVDKA"
    },
    {
      "field": "uses",
      "target": "0670T59HGE"
    },
    {
      "field": "uses",
      "target": "096Z9MP2ME"
    },
    {
      "field": "uses",
      "target": "NS84D2920R"
    },
    {
      "field": "uses",
      "target": "73RYYEVDKA"
    },
    {
      "field": "uses",
      "target": "C31PYYPRGC"
    },
    {
      "field": "uses",
      "target": "XA0DCNH7WY"
    },
    {
      "field": "uses",
      "target": "05QY1VEBYD"
    },
    {
      "field": "uses",
      "target": "F0HQXQ0HCD"
    },
    {
      "field": "uses",
      "target": "30H6B0G2FT"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/NPF42DJMBX.md",
  "msc": [
    "13B35",
    "13C11",
    "13J10"
  ],
  "name": "Completion is flat",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/NPF/NPF42DJMBX.md",
  "status": null,
  "tag": "NPF42DJMBX",
  "uri": "m2://NPF42DJMBX",
  "url": "https://math2.tamnd.com/wiki/completion-is-flat/",
  "writing": {
    "tag": "NPF42DJMBX",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}